【作者】 韩忠杰；

【导师】 许跟起；

【作者基本信息】 天津大学 ， 运筹学与控制论， 2007， 硕士

【摘要】 本研究论文是作者在硕士研究生学习期间参加的一个国家自然基金项目(系列连接弹性振动系统的控制问题)中的一个专题,主要是研究系列连接的Timoshenko弹性梁系统在边界和连接点加控制的条件下,所形成的闭环系统的稳定性与结构。研究在反馈控制之下系列连接弹性梁系统的适定性、渐近稳定性、指数稳定性、谱确定增长条件等性质。对于单根弹性梁振动系统来说,专家学者们常采用乘子办法来得到系统的这些性质,也有些学者采用谱分析方法来做,在这种情况下,系统的特征值往往都比较容易求出,并且性质上也比较简单;但是对于系列连接弹性梁振动系统来说,由于其复杂性,乘子是很难找到的,其特征值也很难求出,并且往往具有多重性、非可分离性等特点,这就需要我们去寻找其它方法来解决这个问题。本文处理系统的最大特点在于把系统微分方程用矩阵形式来描述,把每根梁都放在矩阵中统一地去处理,利用渐近分析的技巧,给出谱的分布区域,然后根据指数型函数的性质,利用谱的分布得到系统本征向量Riesz基的性质,从而得到系统的谱确定增长条件。本文主要得到两方面的结果,一方面,对于一端固定一端自由的系列连接的Timoshenko弹性梁系统,假设在连接点处其横向位移和旋转角度是连续的,而剪切力和弯曲力矩不连续,在此情况下,通过在连接点处和右端点处施加反馈控制器,得到闭环系统的渐近稳定性及(广义)本征向量的Riesz基性质,从而得到系统满足谱确定增长条件,并证明得到在n=3时,闭环系统是指数稳定的。另一方面,对于两端固定的系列连接Timoshenko弹性梁系统,假设在连接点处其剪切力和弯曲力矩是连续的,而横向位移和旋转角度不连续,在此情况下,通过在连接点处施加反馈控制器,并设置补偿器,得到闭环系统的稳定性及(广义)本征向量Riesz基性质,从而得到谱确定增长条件。本文是以系列连接的Timoshenko梁系统为研究对象进行研究的,由于我们采用的方法具有一般性,这样的方法可以推广应用到其他系统模型的研究中,诸如:系列连接的弦系统和Euler-Bernoulli弹性梁系统以及网络结构弹性梁系统等。更多还原

【Abstract】 This report is one part of the research on the control problem of serially connected elastic system which was done by author during postgraduate student studying. The aim is to study the feedback stabilization of the serially connected Timoshenko system and to discuss the system’s properties such as well-posed, asymptotic stability, exponential stability, spectrum determined growth condition e.t.c. In the case of single beam, these properties were usually discussed by using multiplier method which is still an important way to study the system stability. Under this case the eigenvalues usually can be calculated and the properties of them is simple. However, for serially connected system, it is difficult to find a multiplier for the system. The eigenvalues of system are very complex and usually can not be obtained. So another way should be found to solve this problem. Frequency domain method is used to study the stability of serially connected Timoshenko elastic system in this report. The distinguishing feature of this report is that the matrix form is used to denote system, by the technique of asymptotic analysis, the distribution of spectrum is given. Then the Riesz basis property of the eigenvectors and generalized eigenvectors of the operator is proved. So the spectrum determined growth condition holds. By using this method, we not only studied the stability of the closed loop system, but also got the properties of the spectrum distribution and the spectrum determined growth condition.Two results of the serially connected Timoshenko beams can be gotten in this report. On one hand, serially connected Timoshenko beams with joint and boundary feedback controls is studied. Supposed that the left end of whole beam is clamped and the right end is free. At intermediate nodes, the displacement and rotational angle of beams are continuous but the shearing force and bending moment could be discontinuous. The collocated velocity feedback of the beams at intermediate nodes and the right end are used to stabilize the system, then Riesz basis properties of the closed loop system are proved to be true. Hence the spectrum determined growth condition holds. Furthermore, the closed loop system is exponentially stable for the case of n = 3.On another hand, stabilization problem of n-connected Timoshenko beams is discussed. Supposed that both ends of the beams are clamped. At intermediate nodes, the shearing force and bending moment are continuous, but the displacement and rotational angle of beams are discontinuous. Shearing force and bending moment at intermediate nodes are observed. The compensators are signed to use to obtain the displacements and that the closed loop system is asymptotically stable. By a detail spectral analysis of the system, it is shown that the closed loop system is of Riesz basis property under some conditions. Hence the spectrum determined growth condition holds.Although the study of this report is mainly on serially connected Timoshenko elastic system, this method used in this report can be generalized in the study of other models, such as serially connected Euler-Bernoulli elastic system, network configuration elastic system and so on更多还原